Non-standard models of Peano Arithmetic

From FOM Digest, Vol 101, Issue 34,a post by Walt Read:

… the result of the work on non-standard (non-Euclidean) models of geometry was a recognition of the other models as equally valid. Eventually it was considered reasonable that even “the” universe might better be modeled by one of the non-standard models. Do we see N the same way? Is there a unique thing denoted by “the natural numbers”, accessible to us through intuition or however, with the non-standard models of PA being essentially artifacts of the formalization process? Or do the non-standard models have equal claim as models of “the natural numbers”? Might we at some time in the future see one of the non-standard models as better suited to our understanding as we learn more about natural numbers?